# Properties

 Conductor 723 Order 2 Real Yes Primitive Yes Parity Odd Orbit Label 723.b

# Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(723)
sage: chi = H[722]
pari: [g,chi] = znchar(Mod(722,723))

## Kronecker symbol representation

sage: kronecker_character(-723)
pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-723}{\bullet}\right)$$

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 723 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 2 Real = Yes sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 723.b Orbit index = 2

## Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(242,7)$$ → $$(-1,-1)$$

## Values

 -1 1 2 4 5 7 8 10 11 13 14 16 $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 723 }(722,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{723}(722,\cdot)) = \sum_{r\in \Z/723\Z} \chi_{723}(722,r) e\left(\frac{2r}{723}\right) = -26.8886593195i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 723 }(722,·),\chi_{ 723 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{723}(722,\cdot),\chi_{723}(1,\cdot)) = \sum_{r\in \Z/723\Z} \chi_{723}(722,r) \chi_{723}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 723 }(722,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{723}(722,·)) = \sum_{r \in \Z/723\Z} \chi_{723}(722,r) e\left(\frac{1 r + 2 r^{-1}}{723}\right) = -0.0$$